It is well known that for the sum of independent r.v., the strong approximation can be achieved using Yurinskii's coupling:
Theorem (Yurinskii, 1978). Let $\xi_{1},\ldots,\xi_{n}$ be independent random variables with $\mathbb{E}\left[\xi_{i}\right]=0$ for each $i$ and $\beta\equiv\sum_{i}^{n}\mathbb{E}\left[\left|\xi_{i}\right|^{3}\right]$ finite. Let $S=\xi_{1}+\ldots+\xi_{n}.$ For each $\delta>0$ there exists a random variable $T$ with a $\mathcal{N}\left(0,\mathrm{Var}\left(S\right)\right)$ distribution such that $$\mathbb{P}\left(\left|S-T\right|>3\delta\right)\leq C_{0}B\left(1+\left|\log\left(1/B\right)\right|\right)\quad \text{where }B=\beta\delta^{-3},$$ for some universal constant $C_{0}$.
In above theorem, $\xi_{i}$ are independent and are assumed in $\left(\Omega,\mathcal{F},\mathbb{P}\right)$. I wonder if it still holds when the independent condition is replaced by conditional independence.
That is, suppose now $\xi_{i}$ are defined in the probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$, where the probability space have the following structure: $\Omega=\Omega^{\left(0\right)}\times\Omega^{\left(1\right)}$, $\mathcal{F}=\mathcal{F}^{\left(0\right)}\otimes\mathcal{F}^{\left(1\right)}$. The only difference is that $\xi_{i}$ are not naturally independent, instead, we have $\xi_{i}$ are independent conditional on $\mathcal{F}^{\left(0\right)}$. Does the above Gaussian coupling variable still exist? If so, will the variable be independent of $\mathcal{F}^{\left(0\right)}$?
Any comments / counter examples / hints on proof are welcome. Thanks!